Show that $\int_{\mathbb R^n}e^{|x|^{-n}}dx=$ Volume of n-sphere

Question

I'm preparing for a calculus exam, I'd like help in solving this question.

Let $x \in \mathbb R^n$, $|x|={(x_1^2+x_2^2+...+x_n^2)^{\frac{1}{n}}}$,

Show that $$\int_{\mathbb R^n} e^{|x|^{-n}}dx$$ is equal to the volume of the unit sphere of $\mathbb R^n$.

The way to do this in my opinion is write the integral as

$$\int_{\mathbb R^n}e^{\frac{1}{x_1^2+x_2^2+...+x_n^2}}dx$$

And then go to spherical coordinate as shown here http://en.wikipedia.org/wiki/N-sphere#Spherical_coordinates

But I haven't managed to get a definite result.

Would someone please assist?

Answer 1

Hint: if $x=r\omega$, where $r>0$ and $|\omega|=1$ then $dx=r^{N-1}drd\omega$ and $$\int_{\mathbb{R}^N}e^{-|x|^n}dx=\int_{S_1}\int_0^\infty e^{-r^N}r^{N-1}drd\omega,$$

where $S_1$ is the unit sphere.